Convergence of compact-surface beta-zero packing to the planar density

Let $\mathcal{S}$ be a compact Riemann surface, let $U(z,w)$ be its logarithmic monopole, and for a positive real $\beta$ define

$$\rho_{n,\beta}(\mathcal{S})=\inf_{b,z_1,\ldots,z_n}\frac{1}{a(\mathcal{S})}\int_{\mathcal{S}}\big(b\,\mathrm{e}^{\beta U(\cdot,z_1)+\cdots+\beta U(\cdot,z_n)}-1\big)^2\,\mathrm{d} A_{\mathcal{S}},$$

where the infimum is over positive reals $b$ and points $z_1,\ldots,z_n\in\mathcal{S}$. Compact-surface convergence conjecture. For any fixed compact surface $\mathcal{S}$ and any fixed positive real $\beta$,

$$\lim_{n\to+\infty}\rho_{n,\beta}(\mathcal{S})\to\rho_\beta(\mathbb{C}).$$

The conjecture predicts that the limiting average discrepancy on every fixed compact surface agrees with the planar density; the source gives no resolution.